Finite Group Databasebeta

← #38752245

Subgroups — #38752245

Order 25216.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263356C211yesmaximal, characteristic
2C22×Dic788C211yesmaximal, characteristic, complemented
3#27161650C211yesmaximal, characteristic
4#27161556C211yesmaximal, characteristic, complemented
5#27161300C214yesmaximal, complemented
6#27161288C214yesmaximal, complemented
7#27161105C211yesmaximal, characteristic, complemented
8#27161087C211yesmaximal, characteristic, complemented
9#416×C197C211yesmaximal, characteristic
10C22×Dic394C2211yescharacteristic
11C22×Dic394C2211yescharacteristic
12#20263356—24no—
13C2×Dic788—24no—
14C2×Dic788C2214yes—
15C2×#15691348C2212yes—
16C2×#15691346C2212yes—
17C2×#15691342C2211yescharacteristic
18#20263324C2212yes—
19#20263313C2212yes—
20#20263307—22no—
21#20263305—22no—
22C4×Dic394—22no—
23#20263291C2214yes—
24#20263285C2211yescharacteristic
25#20263282—22no—
26#20263255C2214yes—
27#20263249C2214yes—
28C2×#41×C197C2211yescharacteristic
29C22×C8×C197C2211yescharacteristic
30#117×C197C2211yescharacteristic
31#103×C197C2212yes—
32#102×C197C2212yes—
33C2×Dic394—21no—
34C2×Dic394—21no—
35C2×Dic394—22no—
36C2×Dic394—22no—
37C2×Dic394—22no—
38C2×Dic394D414yes—
39C2×Dic394C2312yes—
40C2×Dic394C2312yes—
41C22×Dic197—21no—
42C22×Dic197—21no—
43Dic788—28no—
44#15691348—24no—
45#15691346—24no—
46#15691342—22no—
47C2×#13096802C2312yes—
48C4⋊Dic197D412yes—
49#15691337—24no—
50#15691337—24no—
51C22⋊Dic197—24no—
52C22⋊Dic197D412yes—
53C4×Dic197—24no—
54C4×Dic197C2312yes—
55#41×C197—22no—
56#41×C197—22no—
57C2×Q8×C197C2311yescharacteristic
58C2×D4×C197—21no—
59C2×D4×C197C2311yescharacteristic
60C22×C4×C197—21no—
61C22×C4×C197C2311yescharacteristic
62C2×C8×C197—22no—
63C2×C8×C197C2312yes—
64C4⋊C4×C197C2312yes—
65C22⋊C4×C197—21no—
66C42×C197—21no—
67Dic394—24no—
68Dic394—22no—
69Dic394—22no—
70Dic394—22no—
71Dic394—42no—
72Dic394C2×D414yes—
73C2×Dic197—24no—
74C2×Dic197—22no—
75C2×Dic197—22no—
76C2×Dic197—42no—
77C2×Dic197C2×D414yes—
78#13096802—22no—
79C23×C197—21no—
80C23×C197C2×D411yescharacteristic
81Q8×C197—21no—
82Q8×C197—21no—
83D4×C197—21no—
84D4×C197—21no—
85D4×C197—42no—
86C2×C4×C197—22no—
87C2×C4×C197—21no—
88C2×C4×C197—42no—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197C2×D412yes—
91C2×C4×C197#4112yes—
92C2×C4×C197C2411yescharacteristic
93C8×C197—22no—
94C22×C197—22no—
95C22×C197—41no—
96C22×C197—42no—
97C22×C197#12112yes—
98C22×C197C22×D411yescharacteristic
99C4×C197—41no—
100C4×C197—42no—
101C4×C197#12112yes—
102C4×C197C22×D411yescharacteristic
103C4×C197C2×#4111yescharacteristic
104Dic197—24no—
105Dic197—24no—
106Dic197—42no—
107C2×C197—22no—
108C2×C197—81no—
109C2×C197C2×#12111yescharacteristic
110C2×C197#57812yes—
111C197#304411yescharacteristic, complemented, Sylow
112#3044—1971nomaximal, Sylow
113C2×#137—1971no—
114C22×Q16—1971no—
115#536—1971no—
116#519—1971no—
117#464—1974no—
118#462—1974no—
119#424—1971no—
120#419—1971no—
121#416D19711yescharacteristic
122C2×#41C2×D19711yescharacteristic
123C22×Q8—1971no—
124C22×Q8—1971no—
125#137—3944no—
126C2×Q16—3944no—
127C2×Q16—1972no—
128C2×Q16—1974no—
129C2×SD16—1972no—
130C2×M16—1971no—
131C22×C8C2×D19711yescharacteristic
132C4⋊Q8—1972no—
133#124—1972no—
134#123—3942no—
135C22⋊Q8—3942no—
136C4×Q8—3942no—
137C4×Q8—1974no—
138#117—3942no—
139#117—1971no—
140#117C2×D19711yescharacteristic
141C4⋊C8—1974no—
142#103—1974no—
143#103C2×D19712yes—
144#102C2×D19712yes—
145#41—22no—
146#41—22no—
147C2×Q8—3942no—
148C2×Q8—3942no—
149C2×Q8—3941no—
150C2×Q8—3941no—
151C2×Q8—3942no—
152C2×Q8—1974no—
153C2×Q8—1972no—
154C2×Q8—1972no—
155C2×Q8C22×D19711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D19711yescharacteristic
158C22×C4—21no—
159C22×C4—3941no—
160C22×C4—3941no—
161C22×C4C22×D19711yescharacteristic
162Q16—3948no—
163Q16—3944no—
164SD16—3944no—
165M16—3942no—
166C2×C8—22no—
167C2×C8—1972no—
168C2×C8C22×D19712yes—
169C4⋊C4—3944no—
170C4⋊C4—3944no—
171C4⋊C4—1972no—
172C4⋊C4C22×D19712yes—
173C22⋊C4—21no—
174C22⋊C4—3944no—
175C22⋊C4—1972no—
176C42—21no—
177C42—3944no—
178C42—1972no—
179C23—21no—
180C23D4×D19711yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—7882no—
184Q8—3942no—
185Q8—3942no—
186Q8—3944no—
187Q8—3942no—
188Q8—1974no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—7882no—
196C2×C4—3942no—
197C2×C4—3944no—
198C2×C4—3942no—
199C2×C4—1974no—
200C2×C4D4×D19711yescharacteristic
201C2×C4D4×D19712yes—
202C2×C4#1569135812yes—
203C2×C4C23×D19711yescharacteristic
204C8—22no—
205C8—3942no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2026329912yes—
210C22C2×D4×D19711yescharacteristic
211C4—42no—
212C4—41no—
213C4—7882no—
214C4—3944no—
215C4—3944no—
216C4#2026329912yes—
217C4C2×D4×D19711yescharacteristic
218C4C2×#1569135811yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2026329911yescharacteristic
222C2#2716189312yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#416×C197C2
socleC22×C197C22×D4
prime radicalC2×#41×C197C22
hypercenter#416D197
hypocenterC197#3044
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×Dic788C2semidirect
4. #27161556C2semidirect
5. #27161300C2semidirect
6. #27161288C2semidirect
7. #27161105C2semidirect
8. #27161087C2semidirect
111. C197#3044semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#202633562
C22×Dic7882
#271616502
#271615562
#271613002
#271612882
#271611052
#271610872
#416×C1972
#3044197